Calculus Examples

$\frac{d y}{d x} = \frac{6 x^{2}}{y}$ , $y (- 1) = - 2$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{6 x^{2}}{y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{6 x^{2}}{y}$

Step 1.2.2

Rewrite the expression.

$y \frac{d y}{d x} = 6 x^{2}$

Step 1.3

Rewrite the equation.

$y d y = 6 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int 6 x^{2} d x$

Step 2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \int 6 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = 6 \int x^{2} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{2} y^{2} + C_{1} = 6 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $6 (\frac{1}{3} x^{3} + C_{2})$ as $6 (\frac{1}{3}) x^{3} + C_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = 6 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $6$ and $\frac{1}{3}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{6}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $6$ and $3$ .

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Step 2.3.3.2.2.1

Factor $3$ out of $6$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{3 \cdot 2}{3} x^{3} + C_{2}$

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $3$ out of $3$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{3 \cdot 2}{3 (1)} x^{3} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \frac{3 \cdot 2}{3 \cdot 1} x^{3} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = \frac{2}{1} x^{3} + C_{2}$

Step 2.3.3.2.2.2.4

Divide $2$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = 2 x^{3} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{2} y^{2} = 2 x^{3} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (2 x^{3} + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (2 x^{3} + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (2 x^{3} + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (2 x^{3} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (2 x^{3} + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

$y^{2} = 2 (2 x^{3}) + 2 K$

Step 3.2.2.1.2

Multiply $2$ by $2$ .

$y^{2} = 4 x^{3} + 2 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{4 x^{3} + 2 K}$

Step 3.4

Factor $2$ out of $4 x^{3} + 2 K$ .

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Step 3.4.1

Factor $2$ out of $4 x^{3}$ .

$y = \pm \sqrt{2 (2 x^{3}) + 2 K}$

Step 3.4.2

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (2 x^{3}) + 2 K}$

Step 3.4.3

Factor $2$ out of $2 (2 x^{3}) + 2 K$ .

$y = \pm \sqrt{2 (2 x^{3} + K)}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{2 (2 x^{3} + K)}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{2 (2 x^{3} + K)}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{2 (2 x^{3} + K)}$

$y = - \sqrt{2 (2 x^{3} + K)}$

$y = \sqrt{2 (2 x^{3} + K)}$

$y = - \sqrt{2 (2 x^{3} + K)}$

$y = \sqrt{2 (2 x^{3} + K)}$

$y = - \sqrt{2 (2 x^{3} + K)}$

Step 4

Since $y$ is negative in the initial condition $(- 1, - 2)$ , only consider $y = - \sqrt{2 (2 x^{3} + K)}$ to find the $K$ . Substitute $- 1$ for $x$ and $- 2$ for $y$ .

$- 2 = - \sqrt{2 (2 {(- 1)}^{3} + K)}$

Step 5

Solve for $K$ .

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Step 5.1

Rewrite the equation as $- \sqrt{2 (2 {(- 1)}^{3} + K)} = - 2$ .

$- \sqrt{2 (2 {(- 1)}^{3} + K)} = - 2$

Step 5.2

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \sqrt{2 (2 {(- 1)}^{3} + K)})}^{2} = {(- 2)}^{2}$

Step 5.3

Simplify each side of the equation.

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Step 5.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (2 {(- 1)}^{3} + K)}$ as ${(2 (2 {(- 1)}^{3} + K))}^{\frac{1}{2}}$ .

${(- {(2 (2 {(- 1)}^{3} + K))}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify ${(- {(2 (2 {(- 1)}^{3} + K))}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.1

Simplify each term.

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Step 5.3.2.1.1.1

Raise $- 1$ to the power of $3$ .

${(- {(2 (2 \cdot - 1 + K))}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.3.2.1.1.2

Multiply $2$ by $- 1$ .

${(- {(2 (- 2 + K))}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.3.2.1.2

Simplify by multiplying through.

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Step 5.3.2.1.2.1

Apply the distributive property.

${(- {(2 \cdot - 2 + 2 K)}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.3.2.1.2.2

Simplify the expression.

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Step 5.3.2.1.2.2.1

Multiply $2$ by $- 2$ .

${(- {(- 4 + 2 K)}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.3.2.1.2.2.2

Apply the product rule to $- {(- 4 + 2 K)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(- 4 + 2 K)}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.3.2.1.2.2.3

Raise $- 1$ to the power of $2$ .

$1 {({(- 4 + 2 K)}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.3.2.1.2.2.4

Multiply ${({(- 4 + 2 K)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(- 4 + 2 K)}^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Step 5.3.2.1.2.2.5

Multiply the exponents in ${({(- 4 + 2 K)}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.2.1.2.2.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- 4 + 2 K)}^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Step 5.3.2.1.2.2.5.2

Cancel the common factor of $2$ .

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Step 5.3.2.1.2.2.5.2.1

Cancel the common factor.

${(- 4 + 2 K)}^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Step 5.3.2.1.2.2.5.2.2

Rewrite the expression.

${(- 4 + 2 K)}^{1} = {(- 2)}^{2}$

Step 5.3.2.1.3

Simplify.

$- 4 + 2 K = {(- 2)}^{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Raise $- 2$ to the power of $2$ .

$- 4 + 2 K = 4$

Step 5.4

Solve for $K$ .

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Step 5.4.1

Move all terms not containing $K$ to the right side of the equation.

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Step 5.4.1.1

Add $4$ to both sides of the equation.

$2 K = 4 + 4$

Step 5.4.1.2

Add $4$ and $4$ .

$2 K = 8$

Step 5.4.2

Divide each term in $2 K = 8$ by $2$ and simplify.

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Step 5.4.2.1

Divide each term in $2 K = 8$ by $2$ .

$\frac{2 K}{2} = \frac{8}{2}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{2 K}{2} = \frac{8}{2}$

Step 5.4.2.2.1.2

Divide $K$ by $1$ .

$K = \frac{8}{2}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide $8$ by $2$ .

$K = 4$

Step 6

Substitute $4$ for $K$ in $y = - \sqrt{2 (2 x^{3} + K)}$ and simplify.

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Step 6.1

Substitute $4$ for $K$ .

$y = - \sqrt{2 (2 x^{3} + 4)}$

Step 6.2

Factor $2$ out of $2 x^{3} + 4$ .

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Step 6.2.1

Factor $2$ out of $2 x^{3}$ .

$y = - \sqrt{2 (2 (x^{3}) + 4)}$

Step 6.2.2

Factor $2$ out of $4$ .

$y = - \sqrt{2 (2 x^{3} + 2 \cdot 2)}$

Step 6.2.3

Factor $2$ out of $2 x^{3} + 2 \cdot 2$ .

$y = - \sqrt{2 (2 (x^{3} + 2))}$

$y = - \sqrt{2 \cdot 2 (x^{3} + 2)}$

Step 6.3

Multiply $2$ by $2$ .

$y = - \sqrt{4 (x^{3} + 2)}$

Step 6.4

Rewrite $4$ as $2^{2}$ .

$y = - \sqrt{2^{2} (x^{3} + 2)}$

Step 6.5

Pull terms out from under the radical.

$y = - (2 \sqrt{x^{3} + 2})$

Step 6.6

Multiply $2$ by $- 1$ .

$y = - 2 \sqrt{x^{3} + 2}$